A Simpson correspondence in positive characteristic

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A Simpson correspondence in positive characteristic

Michel Gros, Bernard Le Stum, Adolfo Quirós

To cite this version:

Michel Gros, Bernard Le Stum, Adolfo Quirós. A Simpson correspondence in positive characteristic.

Publications of the Research Institute for Mathematical Sciences, European Mathematical Society, 2010, 46 (1), pp.1-35. �hal-00337672�

A Simpson correspondence in positive characteristic

Michel Gros, Bernard Le Stum & Adolfo Quir´os^∗ November 7, 2008

Contents

1 Usual divided powers 2

2 Higher divided powers 4

3 The p^m-curvature map 6

4 Lifting the p^m-curvature 11

5 Higgs modules 19

6 Informal complements 23

Introduction

The first two authors had the opportunity to participate in a working group in Rennes dedicated to the work of Arthur Ogus and Vadim Vologodsky on non abelian Hodge theory, which is now published in [8]. This is an analog in positive characteristic p of Simpson’s correspondence over the complex numbers between local systems and a certain type of holomorphic vector bundles that he called Higgs bundles ([9]). Actually, Pierre Berthelot had previous results related to these questions and he used this opportunity to explain them to us. What we want to do here is to extend these results to differential operators of higher level. In the future, we wish to lift the theory modulo some power ofpand compare to Faltings-Simpsonp-adic correspondence ([4]).

We would also like to mention some other papers related to our investigation. First, there is an article [6] by Masaharu Kaneda where he proves the (semi-linear) Azumaya nature of the ring of differential operators of higher level, generalizing the result of [3]. Also, Marius van der Put in [10] studies the (linear) Azumaya nature of differential operators in the context of differential fields.

∗A. Q. was partially supported by Project GALAR (MTM2006-10548) from MEC (Spain) and by the joint Madrid Region-UAM project TENU2 (CCG07-UAM/ESP-1814).

We will first recall, in sections 1 and 2, in an informal way the notion of divided powers of higher level and how this leads, using some duality, to arithmetic differential operators.

Then, in section 3, we will define the notion of p^m-curvature and show that Kaneda’s isomorphism still holds over an arbitrary basis. Next, in section 4, we assume that there exists a strong lifting of Frobenius modp²and use it to lift the divided Frobenius and derive a Frobenius map on the ring of differential operators of levelm. Actually, we can do better and prove in Theorem 4.13 that this data determines a splitting of a central completion of this ring. It is then rather formal, in section 5, to obtain a Simpson correspondence and we can even give some explicit formulas. We finish the article, in section 6, with a series of complements concerning compatibility with other theories.

Acknowledgments

Many thanks to Pierre Berthelot who explained us the level zero case and helped us understand some tricky constructions.

Conventions

We letpbe a prime andm∈N. Actually, we are interested in them-th power of p. When m = 0, we have p^m = 1 which is therefore independent of p. We may also consider the casem=∞in which case we will write p^m = 0. Again, this is independent ofp.

Unlessm= 0 or m=∞, all schemes are assumed to beZ_(p)-schemes.

We use standard multiindex notations, hoping that everything will be clear from the context.

1 Usual divided powers

It seems useful to briefly recall here some basic results on usual divided powers that we will need afterwards. There is nothing new but we hope that it makes the next sections easier to read without referring to older articles. The main point in this section is to clarify the duality between divided powers and regular powers.

LetRbe a commutative ring (in a topos).

Definition 1.1 A divided power structure on an ideal I in a commutative R-algebra A is a family of maps

I ^//A f ^//f^[k]

that behave like f 7→ ^f_k!^k. We then say that I is a divided power ideal or that A is a divided powerR-algebra.

We will not list all the required properties. Note however that we always have f^[k]f^[l]=

k+l k

f^[k+l].

Theorem 1.2 The functorA 7→ I from divided powerR-algebras toR-modules has a left adjointM 7→Γ_•M.

Proof : See for example, Theorem 3.9 of [2].

Actually Γ_•Mis a graded algebra with divided power ideal Γ_>0M. Not also that Γ_•Mis generated asR-algebra by all thes^[k]fors∈M. For example, ifMis free on{s_λ, λ∈Λ}, then Γ_kM is free on thes^[k]:=Q

s^[k_λ^λ] with|k|:=P

k_λ =k. Moreover, multiplication is given by the general formula for divided powers recalled above.

Definition 1.3 If M is an R-module, the ring Γ_•Mis called the divided power algebra on M.

IfMis anR-module, we will denote byS^•Mthesymmetric algebra onMand by ˇMthe dual of M. Also, we will denote by S[^•Mthe completion of S^•Malong S^>0M.

Proposition 1.4 If Mis an R-module, there exists a canonical pairing S^•M ס Γ_•M ^//R

(ϕ₁· · ·ϕ_n, s^[n]) ^//ϕ1(s)· · ·ϕn(s)

giving rise to perfect duality at each step when Mis locally free of finite type.

Proof : See for example, Proposition A.10 of [2]).

The general formula for this pairing is quite involved but if{s_λ, λ∈Λ}is a finite basis for M, and{ˇs_λ, λ∈Λ}denotes the dual basis, then the dual basis to{ˇs^k}is nothing else but {s^[k]}.

Corollary 1.5 If Ma locally free R-module of finite type, we have a perfect pairing S[^•M ס Γ_•M ^//R.

Of course, there exists also a natural mapS^•M →Γ_•Mbut it is not injective in general:

ifp^N+1 = 0 onX, then f^pn 7→pⁿf^[pn]= 0 forn > N.

Proposition 1.6 If M is an R-module, multiplication on S^•Mˇ is dual to the diagonal map

Γ_•M ^{δ //}Γ_•(M ⊕ M) ^{≃ //}Γ_•M ⊗Γ_•M

s^{[k] //}P

i+j=ks^[i]⊗s^[j]

Proof : The point is to show that (ϕ₁· · ·ϕ_i⊗ψ₁· · ·ψ_j)◦δ acts like ϕ₁· · ·ϕ_iψ₁· · ·ψ_j on s^[k] whenk=i+j. And this is clear.

2 Higher divided powers

We quickly recall the definition of Berthelot’s divided powers of level m and how one derives the notion of differential operators of higher level from them (see [1] for a detailed exposition). We stick to a geometric situation.

Definition 2.1 LetX ֒→Y be an immersion of schemes defined by an idealI. Adivided power structure of level m on I is a divided power ideal J ⊂ O_Y such that

I^(pm)+pI ⊂ J ⊂ I.

Here,I^(pm) denotes the ideal generated by p^m-th powers of elements of I.

It is then possible to definepartial divided powers onI : they are maps I ^//A

f //f^{k}

that behave likef 7→ ^f_q!^k where q is the integral part of _p^km. Actually, if k=qp^m+r and f ∈ IX, one sets

f^{k}:=f^r(f^pm)^[q].

We have, as above, a multiplication formula (writing q_k instead of q in order to take into account the dependence onk):

f^{k}f^{l} =

k+l k

f^{k+l} where

k+l k

= q_k+l! q_k!q_l!.

When m = 0, we must have J = I and f^{k} = f^[k] is the above divided power. When m=∞, the condition reduces topJ ⊂ I ⊂ J and we may always chooseJ =I sincepI has divided powers. Also, in this case, f^{k} =f^k is just the usual power.

We fix a (formal) schemeSwith a divided power structure of level mon some ideal ofO_S and we assume that all constructions below are made over S and are “compatible” with the divided powers on S in a sense that we do not want to make precise here (see [1] for details). Actually, in the case of a regular immersion, the divided power envelope defined below does not depend onS and this applies in particular to the diagonal embedding of a smooth S-scheme.

Proposition 2.2 If an immersionX ֒→Y has divided powers of levelm, there is a finest (decreasing) ring filtration I^{n} such that I^{1} =I and such that f^{h} ∈ I^{nh} whenever f ∈ I^{n}.

Proof : See Proposition 1.3.7 of [1].

Proposition 2.3 The functor that forgets the divided power structure on an immersion X ֒→Y has a left adjoint X ֒→P_Xm(Y).

Proof : See proposition 1.4.1 of [1].

Definition 2.4 IfX ֒→Y is an immersion of schemes, thenP_Xm(Y)is called the divided power envelope of level m of X in Y.

We will denote by P_Xm(Y) the structural sheaf of P_Xm(Y) and by I_Xm(Y) the divided power ideal of level m. We will also need to consider the usual divided power ideal J_Xm(Y) (in [1], these ideals are denoted byI andIeifI denotes the ideal that defines the immersion). For each n, we will denote by P_Xmⁿ (Y) the subscheme defined by I_Xm^{n+1}(Y) and consider its structural sheaf

P_Xmⁿ (Y) =PXm(Y)/I_Xm^{n+1}(Y).

We will mainly be concerned with diagonal immersionsX ֒→ X×_SX, and we will then write P_Xm,P_Xm,I_Xm,J_Xm,P_Xmⁿ and P_Xmⁿ respectively.

If we are given local coordinatest₁, . . . , t_r on X/S, the ideal I of the diagonal immersion is generated by the τ_i = 1⊗t_i−t_i⊗1. We always implicitly use the first projection as structural map and therefore writet_i⊗1 =t_i and 1⊗t_i =t_i+τ_i. When m= 0, P_Xm is nothing but the divided power algebra on the freeOX-module on the generatorsτ1, . . . , τr. Of course, for m=∞, this is just the symmetric algebra.

In general, we obtain

O_Xhτ₁, . . . , τ_ri^(m):={X

finite

f_iτ^{i}, f_i∈ O_X}

with multiplication given by the general formula for divided powers of level m recalled above.

Definition 2.5 If X is an S-scheme, the dual to P_Xmⁿ is the sheaf D_X,n^(m) of differential operators of level m and order at most n and D_X^(m) =∪_nD^(m)_X,n is the sheaf of differential operators of level m on X/S.

There is a composition law onD^(m)_X that comes by duality from the morphism P_Xm ^δ //P_Xm⊗ P_Xm

a⊗b ^//(a⊗1)⊗(1⊗b).

When X/S is smooth, this turnsD^(m)_X into a non commutative ring.

Locally, we see that

D_X^(m)={X

finite

f_i∂^<i>, f_i∈ O_X}

where∂^<i> is the dual basis toτ^{i} and multiplication on differentials is given by

∂^<k>∂^<l>=

k+l k

∂<k+l> with

k+l k

= k+l

k

nk+l

k

o.

We also have

∂^<k>f =X

i≤k

k i

∂^<i>(f)∂<k−i>.

In this last formula, we implicitly make D_X^(m) act on O_X. This is formally obtained as follows: a differential operator of ordernis nothing but a linear mapP :P_Xmⁿ → OX and we compose it on the left with the map induced by the second projectionp^∗₂:O_X → P_Xmⁿ . For example, if we work locally, thent^h is sent byp^∗₂ to

(t+τ)^h=X

k

h k

t^h−kτ^k =X

k

q_k! h

k

t^h−kτ^{k}

and therefore,

∂^<k>(t^h) =q_k! h

k

t^h−k.

Finally, note that D⁽⁰⁾_X is locally generated by∂₁, . . . , ∂_r and thatD_X^(∞) is Grothendieck’s ring of differential operators. In general, when k < p^m+1, it is convenient to define

∂^[k] = ∂^<k>/q_k! and note that D_X^(m) is locally generated by the ∂_i^[pl] = ∂^<pl> for l ≤ m.

In particular, we see that the diamond brackets notation should not appear very often in practice.

3 The p

-curvature map

We assume from now on that m6=∞.

If X is a scheme of characteristic p, we will denote by F :X → X the m+ 1-st iterate of its Frobenius endomorphism (given by the identity on X and the map f 7→ f^pm+1 on functions).

Lemma 3.1 Let X ֒→Y be an immersion defined by an ideal I. Then, the map I ^//P_Xm(Y)

ϕ //ϕ^{pm+1},

composed with the projection

P_Xm(Y)7→ P_Xm(Y)/IP_Xm(Y), is an F^∗-linear map that is zero on I².

Proof : If ϕ, ψ∈ I, we have

(ϕ+ψ)^{pm+1}=ϕ^{pm+1}+ψ^{pm+1}+ X

i+j=pm+1 i,j>0

p^m+1 i

ϕ^{i}ψ^{j}.

When 0< i, j < p^m+1, we haveq_i, q_j < pandq_i! andq_j! are therefore invertible. It follows that the last part in the sum falls insideI^pm+1. In particular, it is zero modulo IPXm(Y) and it follows that the composite map is additive.

Also, clearly, if f ∈ O_X and if ϕ∈ I, thenf ϕis sent to

(f ϕ)^{pm+1} =f^pm+1ϕ^{pm+1} =F^∗(f)ϕ^{pm+1}.

And we see that the map is F^∗-linear. Finally, ifϕ, ψ ∈ I, thenϕψ is sent to (ϕψ)^{pm+1}=ϕ^pm+1ψ^{pm+1} ∈ IP_Xm(Y).

For the rest of this section, we fix a base schemeS of characteristic pand we assume that X is an S-scheme. We consider the usual commutative diagram with cartesian square (recall that hereF denotes them+ 1-st iteration of Frobenius)

X FX

//((

F

((X^{′ //}

X

S ^{F //}S.

If we apply Lemma 3.1 to the case of the diagonal embedding ofX inX×_SX, we obtain, after linearizing and since F_X^∗Ω¹_X′ =F^∗Ω¹_X, an O_X-linear map

F_X^∗Ω¹_X′ //P_Xm/IP_Xm, that we will calldivided Frobenius.

We may now prove the level m version of Mochizuki’s theorem ([8], Proposition 1.7):

Proposition 3.2 If X is a smooth S-scheme, the divided Frobenius extends uniquely to an isomorphism of O_X-modules

F_X^∗Ω¹_X′/S ≃ I_XmP_Xm^pm+1/IP_Xm^pm+1.

Proof : From the discussion above, it is clear that we have such a map. In order to show that this is an isomorphism, we may assume that there are local coordinates t₁, . . . , t_r on X, pull them back ast^′₁, . . . , t^′_r onX^′ and also set as usualτ_i := 1⊗t_i−t_i⊗1. Then, our map is simply

L_r

i=1O_Xdt^′_i ^≃ //L_r

i=1O_Xτ_i^{pm+1}

dt^′_i ^//τ_i^{pm+1}. Actually, we can do a little better.

Proposition 3.3 If X is a smoothS-scheme, then I_Xm^{pm+1}∩ IP_Xm is stable under usual divided powers. Moreover, the divided Frobenius extends uniquely to an isomorphism of divided power OX-algebras

F_X^∗Γ_•Ω¹_X′/S ≃ P_Xm/IP_Xm.

Proof : The first question is local and we assume for the moment that it is solved. Then, by definition of the divided power algebra on a module, the above map

F_X^∗Ω¹_X′ → P_Xm/IP_Xm extends uniquely to a morphism of divided power algebras

F_X^∗Γ_•Ω¹_X′ = Γ_•F_X^∗Ω¹_X′ → P_Xm/IP_Xm.

Showing that it is an isomorphism is local again.

Thus, we assume that there are local coordinates t₁, . . . , t_r on X, we pull them back as t^′₁, . . . , t^′_r on X^′ and we also set as usualτ_i := 1⊗t_i−t_i⊗1.

We have

P_Xm/IP_Xm=O_X < τ₁, . . . , τ_r >^(m)/(τ₁, . . . , τ_r)

which is therefore a freeO_X-module with basisτ^{kpm+1}. The first assertion easily follows.

Moreover, our map is

O_X <dt^′₁, . . . ,dt^′_r>^{(0) //}O_X < τ₁, . . . , τ_r>^(m)/(τ₁, . . . , τ_r)

dt^′_i ^//τ_i^{pm+1}.

And the left hand side is the free O_X-module with basis dt^′[k]. Our assertion is therefore a consequence of the first part of Lemma 3.4 below.

Lemma 3.4 In an ideal with partial divided powers of level m, we always have 1. For any k∈N,

(f^{pm+1})^[k]= (kp)!

(p!)^kk!f^{kpm+1} and _(p!)^(kp)!kk! ∈1 +pZ.

2. If t=qp^m+r with q < p and r < p^m, then f^{kpm+1}f^{t}=

kp+q q

f^{kpm+1+t}

and ^kp+q_q

∈1 +pZ.

Proof : The first assertion comes from the case m = 0 applied to f^pm. And we may consider the formula

(f^[p])^[k]= (kp)!

(p!)^kk!f^[kp]

as standard. Moreover, there exists a product formula for the factor:

(kp)!

(p!)^kk! =

k−1Y

j=1

jp+p−1 p−1

and it is therefore sufficient to prove that each factor in this product falls into 1 +pZ. We already know that they belong toZand we have the product formula inZ_(p):

jp+p−1 p−1

=

p−1Y

i=1

(1 + j ip).

The second assertion is even easier and comes from f^{u}f^{t} =nu

t

of^{u+t} and

kp^m+1+t t

=

kp+q q

= Yq

i=1

(1 + k ip).

Definition 3.5 If X is a smooth S-scheme, the p^m-curvature map is the morphism F_X^∗S^•T_X^′ → D^(m)_X

obtained by duality from the composite

P_Xm→ P_Xm/IP_Xm≃F_X^∗Γ_•Ω¹_X′

We have to be a little careful here : first of all, we consider the induced morphisms P_Xm^[k] →F_X^∗Γ_≤kΩ¹_X′,

(with usual divided powers onP_Xm), and then we dualize to get F_X^∗S^≤kT_X^′ → D^(m)_X,k.

and take the direct limit on both sides.

Alternatively, we may also call p^m-curvature the adjoint map S^•T_X^′ →F_X∗D^(m)_X .

We will denote byZ_X^(m) the center ofD_X^(m) and byZO_X^(m) the centralizer of O_X inD_X^(m). Proposition 3.6 If X is a smooth scheme over S, then the p^m-curvature map induces an isomorphism of O_X-algebras F_X^∗S^•T_X^′ ≃ ZO_X^(m) and an isomorphism of O_X^′-algebras S^•T_X^′ ≃F_X∗Z_X^(m).

Note that it will formally follow from the definition of the multiplication in D^(m)_X and Proposition 1.6 that the p^m-curvature map is a morphism of algebras. More precisely, this map is obtained by duality from a morphism of coalgebras. However, we need a local description in order to prove the rest of the proposition.

Proof : Both questions are local and we may therefore use local coordinates t₁, . . . , t_r, pull them back tot^′₁, . . . , t^′_r onX^′ and denote byξ^′₁, . . . , ξ_r^′ the corresponding basis ofT_X^′. By construction, thep^m-curvature map is then given by

ξ^′k_i 7→∂_i^<kpm+1>.

It follows from Lemma 3.4 (and duality) that

∂_i^<kpm+1>= (∂_i^<pm+1>)^k

and this shows that we do have a morphism of rings, which is clearly injective because we have free modules on both sides. The image is the O_X^′-subalgebra generated by the

∂^<p_i ^m+1> and this is exactly the center as Berthelot showed in Proposition 2.2.6 of [1].

The following theorem is due to Masaharu Kaneda ([6], section 2.3, see also [3] in the case m= 0) whenS is the spectrum of an algebraically closed field.

Theorem 3.7 Let X be a smooth scheme over a scheme S of positive characteristic p and F_X :X → X^′ the m+ 1-st iterate of the relative Frobenius. Let D_X^(m) be the ring of differential operators of level m on X/S and ZO^(m)_X the centralizer ofOX. Then, there is an isomorphism of ZO^(m)_X -algebras

F_X^∗F_X∗D_X^{(m) //}End

ZO_X^(m)(D_X^(m)) f ⊗Q ^//(P 7→f P Q).

Proof : The question is local and one easily sees that D^(m)_X/S is free asZO_X^(m)-modules on the generators ∂^<k> with k < p^m+1. More precisely, this follows again from Lemma 3.4 that gives us, by duality,

∂_i^<kpm+1+t> = (∂_i^<pm+1>)^k∂_i^<t>

when t < p^m+1. It is then sufficient to compare basis on both sides (see Kaneda’s proof for the details).

For example, when m = 0, in the simplest case of an affine curve X = SpecA with coordinate t and corresponding derivation ∂, the first powers 1, ∂, . . . , ∂^p−1 form a basis of the ring of differential operatorsD and the map of the theorem sends∂^k to

0 ∂^pI_k I_p−k 0

∈Mp×p(A[∂^p]) fork= 0, . . . , p−1, and∂^p to ∂^pI_p.

This theorem is usually stated asproving the Azumaya nature ofF_X∗D_X^(m). More precisely, we can seeF_X∗D^(m)_X as a sheaf of algebras on

Tˇ_X^′ = SpecS^•T_X^′ ≃SpecF_X∗Z_X^(m)

and the above theorem provides a trivialization of FX∗D_X^(m) along the “Frobenius”

F_X :X⊗_X^′ Tˇ_X^′ →Tˇ_X^′.

Proposition 3.8 Let X be a smooth S-scheme. If we denote by K^(m)_X the two-sided ideal of D^(m)_X generated by the image of T_X^′ under the p^m-curvature map, there is an exact sequence

0→ K_X^(m)→ D^(m)_X → End_OX′(O_X)→0.

Proof : This follows from [1], Proposition 2.2.7.

We will denote byDb^(m)_X the completion ofD^(m)_X along the two-sided idealK^(m)_X . We will also denote byZb_X^(m) the completion ofZ_X^(m) alongZ_X^(m)∩ K^(m)_X and byZOb _X^(m) the completion ofZO^(m)_X along ZO_X^(m)∩ K^(m)_X . Note that thep^m-curvature map gives isomorphisms

F_X^∗S\^•T_X^′ ≃ZOb ^(m)_X and S\^•T_X^′ ≃FX∗Zb_X^(m).

Proposition 3.9 If X is a smooth S-scheme, we have natural isomorphisms Zb_X^(m)⊗

Z^(m)_X D^(m)_X ≃Db_X^(m)≃ Hom_OX(P_Xm,O_X).

Proof : The existence of the first map is clear and it formally follows from the definitions that it is an isomorphism.

Now, note that the canonical projectionsP_Xm→ P_Xmⁿ induce a compatible family of maps D_Xn^(m)=Hom_OX(P_Xmⁿ ,O_X)→ Hom_OX(P_Xm,O_X)

from which we derive a morphism D_X^(m) → Hom_OX(P_Xm,O_X). On the other hand, us- ing the isomorphism of Proposition 3.3 and the p^m-curvature, the projection P_Xm → P_Xm/IP_Xm dualizes to

Zb_X^(m) ֒→ HomOX(PXm,OX).

And by construction, these two maps are compatible onZ_X^(m) and induce a map Zb_X^(m)⊗

Z_X^(m) D_X^(m)→ Hom_OX(P_Xm,O_X).

It is now a local question to check that this is an isomorphism.

4 Lifting the p

-curvature

We will prove here the Azumaya nature of the ring of differential operators of higher level. In order to make it easier to read, we will not always mention direct images under Frobenius. This is not very serious because Frobenius maps are homeomorphisms and playing with direct image only impacts the linearity of the maps (and this should be clear from the context).

Definition 4.1 If X is a scheme of characteristic p, a lifting Xe of X modulo p² is a flat Z/p²Z-scheme Xe such that X = Xe ×_Z/p2ZF_p. A lifting of a morphism f :Y → X of schemes of characteristic p is a morphism fe: Xe → Ye between liftings such that f = fe×_Z/p2ZF_p.

We will use the well known elementary result:

Lemma 4.2 If M is aZ/p²Z-module, multiplication by p! induces a surjective map p! :M/pM →pM

which is bijective ifM is flat.

Proof : Exercise.

Note thatp! =−p mod p² and this explains why minus signs will appear in the formulas below. Actually, we will need more fancy estimates :

Lemma 4.3 We have for any m >0, p^m+1

i

=





1 if i= 0 or i=p^m+1 (−1)^kp! if i=kp^m

0 otherwise

mod p².

Proof : Standard results on valuations of factorials show that v_p(

p^m+1 i

) =





0 if i= 0 or i=p^m+1

1 if i=kp^m with 0< k < p

>1 otherwise and we are therefore reduced to showing that

p^m+1 kp^m

= (−1)^kp! mod p², or what is slightly easier, that

p^m+1−1 kp^m−1

= (−1)^k+1 modp.

First of all, we can use Lucas congruences that give p^m+1−1

kp^m−1

=

p−1 k−1

mod p, and then the binomial property

p−1 k−1

+

p−1 k−2

= p

k−2

= 0 modp in order to reduce to the casek= 1.

Up to the end of the section, we let S be a scheme of characteristic pand denote by Sea lifting ofS (modulop² as defined above).

Definition 4.4 IfX is anS-scheme, a strong liftingFe :Xe →Xe^′ of them+ 1-st iteration of Frobenius of X is a morphism that satisfies

f^′ = 1⊗f modp ⇒ Fe^∗(f^′) =f^pm+1+pg^pm with g∈ O_Xe.

When m = 0, this is nothing but a usual lifting but the condition is stronger in general.

For example, the mapt7→t⁴+ 2tis not a strong lifting of the Frobenius on the affine line when m= 1 and p= 2. However, the condition is usually satisfied in practice, especially when the lifting comes from a lifting of the absolute Frobenius as the next lemma shows.

Lemma 4.5 If Fe : Xe → Xe is a lifting of the true absolute Frobenius of X, then for f ∈ O_Xe, we have

Fe^m+1∗(f) =f^pm+1+pg^pm withg∈ O_Xe.

Proof : By definition, we can write

Fe^∗(f) =f^p+pg withg∈ O_Xe and we prove by induction onm that

Fe^m+1∗(f) =f^pm+1+pg^pm.

If we apply the ring homomorphismFe^∗ on both sides of this equality, we get Fe^m+2∗(f) =Fe^∗(f)^pm+1+pFe^∗(g)^pm

= (f^p+pg)^pm+1+p(g^p+ph)^pm =f^pm+2+pg^pm+1.

Now, we fix a smoothS-schemeX and letFe:Xe →Xe^′ be a strong lifting of them+ 1-st iteration of the relative Frobenius of X. We will denote by Xe ×Xe (resp. Xe^′×Xe^′) the fibered product overSeand byIe(resp. Ie^′) the ideal ofXe inXe×Xe (resp. Xe^′ inXe^′×Xe^′).

Lemma 4.6 Assume that Fe^∗(f^′) =f^pm+1+pg^pm with g ∈ O_Xe. Let ϕ= 1⊗f −f ⊗1, ϕ^′ = 1⊗f^′−f^′⊗1 andψ= 1⊗g−g⊗1. Then, the composite map

Fe^∗ :Ie^′ ֒→ O_Xe′×Xe^′ Fe^∗×Fe^∗

−→ O_Xe×Xe → P_X,me .

sends ϕ^′ to

p! ϕ^{pm+1}+

p−1X

k=1

(−1)^kf^(p−k)pmϕ^kpm−ψ^pm

! .

Proof : We have

Fe^∗(ϕ^′) = 1⊗f^pm+1−f^pm+1⊗1 + 1⊗pg^pm−pg^pm⊗1 (recall thatp²= 0 on S)e

=ϕ^pm+1+

p^m+1X−1

i=1

p^m+1 i

f^pm+1−iϕⁱ+pψ^pm. We finish with Lemma 4.3.

Proposition 4.7 There is a well defined map 1

p!Fe^∗ :Ie^′ →pP_X,me ≃ P_X,me /pP_X,me ≃ PX,m

that factors through Ω¹_X′ and takes values inside J_Xm. Moreover, the induced morphism 1

p!Fe^∗: Ω¹_X′ → P_X,m is a lifting of divided Frobenius

Ω¹_X^′ → PX,m/IPX,m.

Proof : It follows from lemma 4.6 that the map is well defined: more precisely, we need to check thatFe^∗ sends Ie^′ insidepP_X,me . By linearity, it is sufficient to consider the action on sections ϕ^′ as in the lemma.

Now, sinceFe^∗ is a morphism of rings that sendsIe^′ to zero modulop, it is clear that _p!¹Fe^∗ will sendIe^′2 to 0. Thus, it factors through Ω¹_e

X^′. Actually, since the target is killed byp, it even factors through Ω¹_X′. And it falls insideJ_Xm thanks to the first part.

Finally, the last assertion follows again from the explicit description of the map.

Warning : The quotient map P_X,m → P_X,m/IP_X,m isnot compatible with the divided power structures : τ_i^pm is sent to 0 but (τ_i^pm)^[p]=τ_i^{pm+1} is not.

Proposition 4.8 The divided Frobenius _p!¹Fe^∗ extends canonically to a morphism F_X^∗Γ_•Ω¹_X′ → P_X,m.

By duality, we get a morphism ofO_X-modules

Φ_X :Db^(m)_X −→ZOb ^(m)_X ֒→Db^(m)_X .

Proof : We saw in Proposition 4.7 that the morphism _p!¹Fe^∗ takes values into J_X,m and therefore extends to a morphism of divided power algebras

Γ_•Ω¹_X′ → P_X,m

that we can linearize. Moreover, we saw in Proposition 3.4 that Db^(m)_X ≃ Hom_OX(P_Xm,O_X) and we also have

ZOb ^(m)_X ≃F_X^∗S\^•T_X^′ ≃F_X^∗Hom_OX′(Γ_•Ω¹_X′,O_X^′)≃ Hom_OX(F_X^∗Γ_•Ω¹_X′,O_X).

Definition 4.9 The morphism ΦX is called the Frobeniusof Db_X^(m).

We will simply write Φ when X is understood from the context but we might also use Φ^(m)_X to indicate the level. Note that Φ actually depends on the choice of the strong lifting Fe of F_X.

Proposition 4.10 If we are given local coordinates t₁, . . . , t_r, then

Φ(∂^hni) =







1 if |n|= 0

0 if 0<|n|< p^m

1 p!

P_r

j=1∂_i^[pm](Fe^∗( ˜t^′_j))∂_j^hpm+1i if n=p^m1_i.

Actually, if we have Fe^∗(te^′_j) =te_j^pm+1+pge_j^pm, the third expression can be rewritten

−t^(p−1)p_i ^m∂^hp_i ^m+1i− Xr j=1

∂_i(g_j)^pm∂_j^hpm+1i.

Proof : The point consists in writing Φ(∂^hni) in the topological O_X-basis ∂^hkpm+1i of ZOb ^(m)_X when |n| ≤p^m. By duality, the coefficient of∂^hkpm+1i in Φ(∂^hni) is identical to the coefficient of τ^{n} in the image of 1⊗(dt)^[k] under the morphism

O_X⊗_OX′ Γ_•Ω¹_X′ → P_X,m. Recall that this is exactly

Fe^∗ p!(τe^′)

![k]

.

Since we consider only the case |n| ≤ p^m we may work modulo I^{pm+1} on the right. If we writeFe^∗(te^′_j) =te_j^pm+1+pge_j^pm, we obtain

1

p!Fe^∗(eτ_j^′) = 1 p!

X

s6=0

∂^hsi(Fe^∗(et^′_j))τ^{s}

=−t^(p−1)p_j ^mτ_j^pm− Xr i=1

∂_i(g_j)^pmτ_i^pm modI^{pm+1}.

Thus, we see that the only contributions will come from the case |k| ≤ 1 and that, when k6= 0, the coefficient of τ^{n} is zero unlessn=p^m1_i. Then, there are two cases, firsti6=j in which case, only

1

p!∂_i^[pm](Fe^∗( ˜t^′_j)) =−∂_i(g_j)^pm is left and the case i=j where we obtain

1

p!∂_i^[pm](Fe^∗(˜t^′_i)) =−t^(p−1)p_i ^m−∂_i(g_i)^pm.

For example, whenm = 0, in the case of the affine line with parameter t and derivation

∂ , if we choose the usual lifting of Frobenius t 7→ t^p, we obtain the simple formula Φ(∂) =−t^p−1∂^p.

Formulas are a lot more complicated in general but they become surprisingly nice when we stick to the usual generators of the center.

Proposition 4.11 For all i= 1,· · · , r, we have

Φ(∂_i^hpm+1i) =∂_i^hpm+1i+ Φ(∂^hp_i ^mi)^p.

Proof : As above, the coefficient of∂^hkpm+1i in Φ(∂_i^hpm+1i) is identical to the coefficient of τ_i^{pm+1} in

Fe^∗

p!(τe^′)^[k]= Yr j=1

τ_j^{pm+1}+

p−1X

l=1

(1)^lt^(p−l)p_j ^mτ_j^lpm− Xr

l=1

∂_l(g_j)^pmτ_l^pm

![kj]

if we write Fe^∗(te^′_j) =te_j^pm+1+pge_j^pm.

Thus, we see that the only contributions will come from the casesk=1_ithat givesτ_i^{pm+1} and k=p1j that will give

−∂_i(g_j)^pmτ_i^pm[p]

= −∂_i(g_j)^pmp

τ_i^{pm+1} for all j plus the special contribution

−t^(p−1)p_i ^mτ_i^pm[p]

=

−t^(p−1)p_i ^mp

τ_i^{pm+1}

of the casej=i. In other words, we obtain Φ(∂_i^hpm+1i) =∂_i^hpm+1i+

−t^(p−1)p_i ^mp

∂_i^hpm+2i+ Xr

j=1

−∂_i(g_j)^pmp

∂_j^hpm+2i.

Our assertion therefore follows from the formulas in Proposition 4.10 because, thanks to

Lemma 3.4, we have

∂^hp_j ^m+1ip

=∂_j^hpm+2i.

This calculation shows in particular that Φ is not a morphism of rings. However, we will see later on that the map induced by Φ on the center is a morphism of rings so that the above formula fully describes this map.

We recall now the following general result on Frobenius and divided powers:

Lemma 4.12 The canonical map P_Xm→ X×X factors through X×_X^′X if X is seen as an X^′-scheme via FX. Actually, if we are given local coordinates t1, . . . , tr and we set τ_i := 1⊗t_i−t_i⊗1, the corresponding map on sections is the canonical injection

O_X[τ]/(τ^pm+1)֒→ O_Xhτi^(m).

Proof : In order to prove the first assertion, the point is to check that if f ∈ OX, then 1⊗f^pm+1−f^pm+1⊗1 is sent to zero in P_Xm. But we have

1⊗f^pm+1−f^pm+1⊗1 =p!(1⊗f−f⊗1)^{pm+1} = 0.

Concerning the second assertion, we just have to verify that O_X[τ]/(τ^pm+1)≃ O_X×X′X.

Since Frobenius is cartesian on ´etale maps ([5], XIV, 1, Proposition 2), we may assume thatX =A^r_S, in which case this is clear.

Our main theorem arrives now:

Theorem 4.13 Let X be a smooth scheme over a scheme S of positive characteristic p and Fe a strong lifting of the m+ 1-st iterate of the relative Frobenius of X. The divided Frobenius extends canonically to an isomorphism

OX×_X′X ⊗O_X′ Γ•Ω¹_X′ ≃ PX,m. By duality, we obtain an isomorphism of O_X-algebras

Db_X^(m)≃ End_b

Z_X^(m)(ZOb _X^(m)).

Proof : Thanks to lemma 4.12, we may extend by linearity the morphism of divided power algebras

Γ_•Ω¹_X′ → P_X,m and obtain

O_X×X′X⊗_OX′ Γ_•Ω¹_X′ → P_X,m.

We now show that it is an isomorphism. This is a local question and we may therefore fix some coordinates t₁, . . . , t_r, call t^′₁, . . . , t^′_r the corresponding coordinates onX^′ and as usual, setτ_i := 1⊗t_i−t_i⊗1. It follows from proposition 4.7 that dt^′_i is sent toτ_i^{pm+1}+φ_i with

φ_i ∈(τ^(pm))O_X[τ]/(τ^pm+1).

And we have to check that the divided power morphism of O_X[τ]/(τ^pm+1)-algebras O_X[τ]/(τ^pm+1)hdt^′i^{(0) //}O_Xhτi^(m)

dt^′_i //τ_i^{pm+1}+φ_i.

is bijective. This is easy because we have free modules with explicit basis on both sides.

Now, we obtain our assertion by duality. Using the fact that X is finite flat over X^′, so thatO_X is locally free over O_X^′, we have the following sequence of isomorphisms

Hom_OX(O_X×X′X⊗_OX′ Γ_•Ω¹_X′,O_X)≃ Hom_OX′(Γ_•Ω¹_X′,Hom_OX(O_X×X′X,O_X))

≃ Hom_OX′(Γ_•Ω¹_X′,End_OX′(O_X))≃ Hom_OX′(Γ_•Ω¹_X′,O_X^′)⊗_OX′ End_OX′(O_X)

≃S\^•T_X^′ ⊗O_X′ EndO_X′(OX)≃ End_S\•T_X′(OX ⊗O_X′ S\^•T_X^′).

and we know thatS^•T_X^′ ≃F_X∗Z_X^(m).

It remains to show that this is a morphism of rings and we do that by proving that it comes by duality from a morphism of coalgebras. Actually, both morphisms

OX×_X′X → PX,m and Γ•Ω¹_X^′ → PX,m

are compatible with the coalgebra structures. We can be more precise: for the first one, this is because the comultiplication is induced on both sides by the same formula

f ⊗g7→f⊗1⊗g

and for the second one, it is an immediate consequence of the universal property of divided powers.

Warning: The isomorphism of the theorem isnot a morphism ofZb_X^(m)-algebras. However, we have the following:

Corollary 4.14 The morphism Φ induces an automorphism of the ring Zb_X^(m) (that de- pends on the lifting of Frobenius).

Proof : The isomorphism of rings

Db_X^(m)≃ End_Zb(m) X

(ZOb _X^(m)).

induces an isomorphism on the centers which is nothing but the map induced by Φ.